Relativistic Motion of a Charged Particle and Pauli Algebra

An element of Pauli algebra consists of a complex scalar, say A, and a three-dimensional complex vector a, denoted A a HA, aL, which thus has eight real dimensions. In effect, this is a generalization of quaternion algebra, but with complex instead of real components. In this article, we call these elements “spinors.” A product of two spinors is a spinor defined by (1) HA, aLÄ⊗ HB, bL = HA B+ a ÿ b , A a+ B a+ i aäbL, where · and ä are the dot and cross products, respectively. Note that this multiplication is associative, implying that we do not need parentheses when multiplying three or more spinors. But multiplication is not commutative (the result depends on the order of factors). There are two important unary (single-argument) operations on spinors: the first is called a reflection (denoted A-), which changes the sign of the vector part, that is, Aa HA, -aL; the second takes the complex conjugate of A and of each component of a; it is denoted A*. Finally, just for convenience, we let A+ denote the combination of both of these, that is, HA-L* = HA*L-. Note that